Quadratic Formula Calculator

Write the equation in standard form ax² + bx + c = 0, identify the three coefficients, and enter them into the a, b and c fields above. Click Calculate Roots and the quadratic formula calculator substitutes your values into x = (-b ± √(b² - 4ac)) / (2a) automatically, returning both roots, the discriminant and the vertex. On a scientific calculator you would do the same by hand: compute b² - 4ac first, take the square root, then evaluate the + and − branches separately.

The discriminant is the expression d = b² - 4ac that appears under the square root in the quadratic formula. It discriminates between the three root cases: d > 0 gives two distinct real roots, d = 0 gives one repeated real root, and d < 0 gives two complex conjugate roots. It also has geometric meaning — it measures how many times the parabola y = ax² + bx + c crosses or touches the x-axis.

A negative discriminant means the square root √(b² - 4ac) is not a real number, so the equation has no real roots. The solutions are a complex conjugate pair of the form p ± qi, where p = -b/(2a) is the real part and q = √(4ac - b²)/(2a) is the imaginary part. The calculator displays these roots, for example 1.5 + 1.3229i and 1.5 - 1.3229i. Geometrically, the parabola never touches the x-axis.

If a = 0, the x² term disappears and the equation becomes linear: bx + c = 0. The quadratic formula divides by 2a, which would be division by zero, so it is undefined. A linear equation has the single root x = -c/b instead. The calculator detects a = 0 and prompts you to correct the input rather than producing a meaningless result.

The vertex lies at x = -b / (2a); substitute that value back to get the y-coordinate, equivalently y = c - b² / (4a). The calculator reports both coordinates automatically. For x² - 3x + 2 the vertex is (1.5, -0.25) — a minimum, since a = 1 > 0. The vertex is also the midpoint between the two real roots, which makes it a quick sanity check on your solution.

Compute the discriminant first. If it is a perfect square (1, 4, 9, 25, ...), the equation factors neatly and factoring is fastest. If the leading coefficient is 1 and the numbers are simple, completing the square is instructive. The quadratic formula always works regardless of factorability, so under exam pressure or with awkward decimals it is the reliable default. All three methods produce identical roots.

Roots, the discriminant and the vertex are computed in full floating-point precision and displayed rounded to 4 decimal places, which exceeds the 2-3 decimal places usually required in coursework. For exact answers, read the substitution steps and restate the root in radical form, e.g. x = (3 + √5) / 2 rather than 2.6180. Very large coefficients or a discriminant extremely close to zero can amplify rounding in the last decimal place.